The Construction And Solution Of Q-mKP Hierarchy With Self-consistent Sources

By introducing a new time flow Ï„k, a new extended q-mKP (q-deformed mKP)hierarchy, called q-mKP hierarchy with self-consistent sources (q-mKPHSCSs), is con-structed under the commutativity ofÏ„k-flow and tn-flow (n k). The first type of q-mKPequation with self-consistent sources is given as the first non-trivial equation in the q-mKPHSCSs, and it reduces to the first type of mKP equation with self-consistent sourceswhen qâ†'1,u≡1. Moreover, the solution of q-mKPHSCSs is obtained by the gaugetransformation between q-KP hierarchy with self-consistent sources (q-KPHSCSs) andq-mKPHSCSs. From the Wronskian solution of q-KPHSCSs, the Wronskian solution ofq-mKPHSCSs is derived. At the end of this paper, the construction of super-symmetricKP hierarchy with self-consistent sources (SKPHSCSs) is discussed.